Valley and spin splittings in PbSe nanowires

We use an empirical tight-binding approach to calculate electron and hole states in [111]-grown PbSe nanowires. We show that the valley-orbit and spin-orbit splittings are very sensitive to the atomic arrangement within the nanowire elementary cell and differ for [111]-nanowires with microscopic $D_{3d}$, $C_{2h}$ and $D_3$ symmetries. For the nanowire diameter below 4 nm the valley-orbit splittings become comparable with the confinement energies and the $\mathit{k}\cdot \mathit{p}$ method is inapplicable. Nanowires with the $D_3$ point symmetry having no inversion center exhibit giant spin splitting $E=\alpha k_z$, linear in one-dimensional wave vector $k_z$, with the constant $\alpha$ up to 1 eV$\cdot$\AA.